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A120503
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Generalized meta-Fibonacci sequence a(n) with parameters s=0 and k=3.
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8
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1, 2, 3, 3, 4, 5, 6, 6, 7, 8, 9, 9, 9, 10, 11, 12, 12, 13, 14, 15, 15, 16, 17, 18, 18, 18, 19, 20, 21, 21, 22, 23, 24, 24, 25, 26, 27, 27, 27, 27, 28, 29, 30, 30, 31, 32, 33, 33, 34, 35, 36, 36, 36, 37, 38, 39, 39, 40, 41, 42, 42, 43, 44, 45, 45, 45, 46, 47, 48, 48, 49, 50, 51, 51
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OFFSET
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1,2
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REFERENCES
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Callaghan, Joseph, John J. Chew III, and Stephen M. Tanny. "On the behavior of a family of meta-Fibonacci sequences." SIAM Journal on Discrete Mathematics 18.4 (2005): 794-824. See T_{0,3} with initial values 0,0,1, and plotted in Fig. 1.5. This is essentially the same sequence. - N. J. A. Sloane, Apr 16 2014
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LINKS
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FORMULA
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If n = 1, a(n)=1. If 2 <= n <= 3, then a(n)=n. If n>3 then a(n)=a(n-a(n-1)) + a(n-1-a(n-2)) + a(n-2-a(n-3))
G.f.: A(z) = z / (1 - z) * prod( (1 - z^(3 * [i])) / (1 - z^[i]), i=1..infinity), where [i] = (3^i - 1) / 2.
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MAPLE
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a := proc(n)
option remember;
if n <= 1 then return 1 end if;
if n <= 3 then return n end if;
return add(a(n - i + 1 - a(n - i)), i = 1 .. 3)
end proc
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MATHEMATICA
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a[n_] := a[n] = If[1 <= n <= 3, n, Sum[a[n-i+1 - a[n-i]], {i, 1, 3}]];
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PROG
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(PARI) {a(n)=local(A); if(n<=3, max(0, n), A=vector(n, i, i); for(k=4, n, A[k]=A[k-A[k-1]]+A[k-1-A[k-2]]+A[k-2-A[k-3]]); A[n])} /* Michael Somos, Aug 31 2006 */
(PARI) apply( A120503(n)={my(s=sumdigits(n*=2, 3)\2); n\=3; while(s>0, s-=valuation(n++, 3)+1); n}, [1..99]) \\ M. F. Hasler, Dec 27 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Frank Ruskey and Chris Deugau (deugaucj(AT)uvic.ca), Jun 20 2006
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STATUS
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approved
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